M.Sc Thesis | |

M.Sc Student | Abu Hamed Mohammad |
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Subject | Frobenius-Schur Indicators for the Representation Categories of Semisimple Lie Algebras |

Department | Department of Mathematics |

Supervisor | PROF. Shlomo Gelaki |

**Let V be a finite dimensional
irreducible representation of a complex semisimple Lie algebra ***g* **and
let m 2 be an integer. The m-th
Frobenius-Schur indicator of V is the number ****,
where is the cyclic automorphism of . In this thesis we derive an
explicit formula for the Frobenius-Schur indicators of V. Our formula is given
in terms of the dimensions of the weight spaces of V, the roots of ***g***
and the Weyl group. As a result we obtain that when m is large enough the
Frobenius-Schur indicators of degree m are all equal to the dimension of the
zero weight space of V. Next we introduce some properties of the 2-nd
Frobenius-Schur indicator of V which are
analogous to the classical case of the representation category of a finite group.
Namely we show that if and only if V is
not self dual and if V is self dual, then if
and only if V admits a nondegenerate bilinear symmetric ***g***
-invariant form and if and only if V
admits a nondegenerate bilinear skew-symmetric ***g*** -invariant form.
In addition we prove Tits theorem which gives an explicit formula for . Finally we calculate the
Frobenius-Schur indicators for the irreducible representations of . In particular we give explicit
formulas for those irreducible representations whose highest weight belongs to
the root lattice of ***g***.**